Questions and Answers of Probability And Stochastic Modeling

Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(\tau\) a stopping time. Find \(\left\langle B^{\tau}ightangle_{t}\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f, g \in \mathcal{L}_{T}^{2}\). Show thata) \(\mathbb{E}\left(f \cdot B_{t} \cdot g \cdot B_{t} \mid
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be \(\mathrm{BM}^{1}\) and \(\left(f_{n}ight)_{n \geqslant 1}\) a sequence in \(\mathcal{L}_{T}^{2}\) such that \(f_{n} ightarrow f\) in
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f \in L^{2}\left(\lambda_{T} \otimes \mathbb{P}ight.\) ) for some \(T>0\). Assume that the limit \(\lim _{\epsilon ightarrow 0}
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1},\left(Y_{t}ight)_{t \geqslant 0}\) and adapted process such that \(Y_{t} Y_{s}=Y_{t}\) for all \(s \leqslant t\) and
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \((0, \infty) i s \mapsto f(s, \omega):=\mathbb{1}_{(0, \infty)}\left(B_{s}ight)\) is continuous in \(L^{2}(\mathbb{P})\).
Let \(\left(X_{n}ight)_{n \geqslant 0}\) be a sequence of r. v. in \(L^{2}(\mathbb{P})\). Show that \(L^{2}-\lim _{n ightarrow \infty} X_{n}=X\) implies \(L^{1}-\lim _{n ightarrow \infty}
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(\int_{0}^{T} B_{s}^{2} d B_{s}=\frac{1}{3} B_{T}^{3}-\int_{0}^{T} B_{s} d s, T>0\).
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion, \(T
Show that \[\mathscr{P}=\left\{\Gamma: \Gamma \subset[0, T] \times \Omega, \Gamma \cap([0, t] \times \Omega) \in \mathscr{B}[0, t] \otimes \mathscr{F}_{t} \quad \text { for all } t \leqslant
Show that every right or left continuous adapted process is \(\mathscr{P}\) measurable.
Let \(T
Let \(\tau\) be a stopping time. Show that \(\mathbb{1}_{[0, \tau)}\) is \(\mathscr{P}\) measurable.
If \(\sigma_{n}\) is localizing for a local martingale, so is \(\tau_{n}:=\sigma_{n} \wedge n\).
The factor \(\mathbb{1}_{\left\{\sigma_{n}>0ight\}}\) is used in the definition of a local martingale, in order to avoid integrability requirements for \(M_{0}\). More precisely, if
Let \(\left(I_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a continuous adapted process with values in \([0, \infty)\) and a.s. increasing sample paths. Set for \(u \geqslant
Extend Theorem 17.1 to local martingales with continuous paths.Data From Theorem 17.1 17.1 Theorem (Doob-Meyer decomposition. Meyer 1962, 1963). For all square integ. Ex. 17.1 rable martingales Me M
For \(M, N \in \mathcal{M}_{T, \text { loc }}^{c}\) we define the covariation by \(\langle M, Nangle_{t}:=\frac{1}{4}\left(\langle M+Nangle_{t}-\langle M-Nangle_{t}ight)\). Show thata) \(t
Use Lemma 17.14 to show the following assertions:a) \(\lim _{n} \mathbb{E}\left(\sup _{t \leqslant T}\left|X_{t}^{n}-X_{t}ight|^{2}ight)=0 \Longrightarrow X^{n} \xrightarrow[n ightarrow \infty]{\text
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\). Show that the following maps are continuous, if we use ucpconvergence in both range and domain:a) \(\mathcal{L}_{T, \text { loc }}^{2}(M) i f \mapsto
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\) and \(f(t, \omega)\) be an adapted càdlàg (right continuous, finite left limits) process and \(t \in[0, T]\).a) Show that \(f \in
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\) and \(f_{n}\) be a sequence in \(\mathcal{L}_{T}^{0}(M)\). If \(f_{n} ightarrow f\) in ucp-sense, then \(f \in \mathcal{L}_{T}^{0}(M)\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be \(B M^{1}\). Use Itô's formula to obtain representations of\[X_{t}=\int_{0}^{t} \exp \left(B_{s}ight) d B_{s} \quad \text { and } \quad Y_{t}=\int_{0}^{t}
Let \(\Pi=\left\{0=t_{0}
Give a direct proof for the identity \[\begin{aligned}\mathbb{E} & {\left[\left(\sum_{l=1}^{N}
Prove the following time-dependent version of Itô's formula: let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f:[0, \infty) \times \mathbb{R} ightarrow \mathbb{R}\) be a
Prove Theorem 5.6 using Itô's formula.Data From Theorem 5.6 5.6 Theorem. Let (Bt, Ft)to be a d-dimensional Brownian motion and assume that the Ex. 5.10 function fee.2 ((0, co) x Rd, R) ne([0, o) x
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Itô's formula to verify that the following processes are martingales:\[X_{t}=e^{t / 2} \cos B_{t} \quad \text
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and set \(r_{t}:=\left|B_{t}ight|=\sqrt{b_{t}^{2}+\beta_{t}^{2}}\).a) Show that the stochastic integrals
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and \(f(x+i y)=u(x, y)+i v(x, y), x, y \in \mathbb{R}\), an analytic function. If \(u_{x}^{2}+u_{y}^{2}=1\), then
Show that the \(d\)-dimensional Itô formula remains valid if we replace the real-valued function \(f: \mathbb{R}^{d} ightarrow \mathbb{R}\) by a complex function \(f=u+i v: \mathbb{R}^{d} ightarrow
Let \(\chi \in \mathcal{C}_{c}^{\infty}(-1,1)\) with \(0 \leqslant \chi \leqslant 1, \chi(0)=1\) and \(\int \chi(x) d x=1\). Set \(\chi_{n}(x):=n \chi(n x)\).a) Show that supp \(\chi_{n} \subset[-1 /
Let \(\sigma_{\Pi}, b_{\Pi} \in \mathcal{S}_{T}\) which approximate \(\sigma, b \in \mathcal{L}_{T}^{2}\) as \(|\Pi| ightarrow 0\), and consider the Itô processes \(X_{t}^{\Pi}=X_{0}+\int_{0}^{t}
Let \(X_{t}=M_{t}+A_{t}\) and \(X_{t}=M_{t}^{\prime}+A_{t}^{\prime}\) where \(M, M^{\prime}\) are continuous local martingales and \(A, A^{\prime}\) are continuous processes which have locally a.s.
State and prove the \(d\)-dimensional version of Lemma 19.1.Data From Lemma 19.1 19.1 Lemma. Let (Bt, Ft)o be a BM, f L (ATP) for all T> 0, and assume that \f(s, w) < C for some C > 0 and all s > 0
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion, i.e. \(\left(b_{t}ight)_{t \geqslant 0},\left(\beta_{t}ight)_{t \geqslant 0}\) are independent
Let \((B(t))_{t \geqslant 0}\) be a \(B M^{1}\) and \(T
Let \((\mathcal{L},\langle\cdot, \cdotangle)\) be a Hilbert space and \(\mathcal{H}\) some linear subspace. Show that \(\mathcal{H}\) is a dense subset of \(\mathcal{L}\) if, and only if, the
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}, n \geqslant 1,0 \leqslant t_{1}
Assume that \(\left(B_{t}, \mathscr{G}_{t}ight)_{t \geqslant 0}\) and \(\left(M_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) are independent martingales. Show that \(\left(M_{t}ight)_{t \leqslant T}\)
Show the following addition to Lemma 19.16:\[\tau(t-)=\inf \{s \geqslant 0: a(s) \geqslant t\} \quad \text { and } \quad a(t-)=\inf \{s \geqslant 0: \tau(s) \geqslant t\}\]and \(\tau(s) \geqslant t
Let \(f:[0, \infty) ightarrow \mathbb{R}\) be absolutely continuous. Show that \(\int f^{a-1} d f=f^{a} / a\) for all \(a>0\).
Use the strong Markov property to show the stationary and independent increments property in Lemma 19.32.c) directly, i.e. for all \(0=u_{0}Data From Lemma 19.32 19.32 Lemma. Let (Bt)to be a BM, Lo
Show that the Hermite Polynomials are linearly independent.
Let \((X, Y)\) be a mean zero Gaussian random variable. Show that \(\mathbb{E}\left(X^{2} Yight)=0\). Use this observation to show that \(I^{2}(f)-\|f\|_{L^{2}}^{2}\) is and
Pick any orthonormal basis \(\left(e_{k}ight)_{k \geqslant 1}\) of \(H\). Let \(\alpha \in \mathbb{N}_{0}^{(\infty)}\), i.e. \(\alpha=\left(\alpha_{1}, \ldots, \alpha_{N}, 0,0, \ldotsight)\) where
Let \(F, G\) be smooth, bounded cylindrical functions. Show thata) \(D_{t}(F G)=G D_{t} F+F D_{t} G\).b) \(D_{t}^{m}(F G)=\sum_{k=0}^{m}\left(\begin{array}{c}m \\ k\end{array}ight) D_{t}^{k} F \cdot
(Special case of Problem 9) Use Itô's formula to show that for any two symmetric f, g L (R)
Let \(F=\exp \left[I_{1}(g)-\frac{1}{2} \int_{0}^{\infty} g^{2}(s) d sight]\) for some \(g \in L^{2}\left(\mathbb{R}_{+}ight)\). Find all derivatives \(D^{k} F\) as well as the Wiener-Itô chaos
Find the Wiener-Itô chaos decomposition for the following random variables ( \(T>0\) is fixed):a) \(\exp \left(B_{T}-T / 2ight)\),b) \(B_{T}^{5}\),c) \(\int_{0}^{1}\left(r^{3} B_{t}^{3}+2 t
Find the Wiener-Itô chaos decomposition for the following random variables ( \(T>0\) is fixed):a) \(\sinh \left(B_{T}ight)\)b) \(\cosh \left(B_{T}ight)\).
Let \(\mathcal{D} \subset L^{2}(\mu)\) and write \(\Sigma:=\operatorname{span}(\mathcal{D})\). Show that \(\Sigma\) is dense if, and only if \(\mathcal{D}\) is total, i.e. \[\bar{\Sigma}=L^{2}(\mu)
Let \((B(t))_{t \in[0,1]}\) be a \({B M^{1}}^{1}\) and set \(t_{k}=k / 2^{n}\) for \(k=0,1, \ldots, 2^{n}\) and \(\Delta t=2^{-n}\). Consider the difference equation\[\Delta
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). The solution of the \(\mathrm{SDE}\)\[d X_{t}=-\beta X_{t} d t+\sigma d B_{t}, \quad X_{0}=\xi\]with \(\beta>0\) and \(\sigma \in
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find an SDE which has \(X_{t}=t B_{t}, t \geqslant 0\), as unique solution.
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find for each of the following processes an SDE which is solved by it:(a) \(U_{t}=\frac{B_{t}}{1+t}\);(b) \(V_{t}=\sin B_{t}\);(c)
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Solve the following SDEsa) \(d X_{t}=b d t+\sigma X_{t} d B_{t}, X_{0}=x_{0}\) and \(b, \sigma \in \mathbb{R}\);b) \(d
Show for all \(x, y \in \mathbb{R}^{n}\) the inequality\[|x-y|(1+|x|)^{-1}(1+|y|)^{-1} \leqslant\left.|| xight|^{-2} x-|y|^{-2} y \mid .\]Use a direct calculation after squaring both sides.
Letbe two globally Lipschitz continuous functions. If \(f\) is bounded, then \(h(x):=f(x) g(x)\) is locally Lipschitz continuous and at most linearly growing. fg: R-R
Let \(A: \mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{d}ight) ightarrow \mathcal{C}_{\infty}\left(\mathbb{R}^{d}ight)\) be a closed operator such that \(\|A u\|_{\infty} \leqslant C\|u\|_{(2)}\); here
Let \(L=L\left(x, D_{x}ight)=\sum_{i j} a_{i j}(x) \partial_{i} \partial_{j}+\sum_{j} b_{j}(x) \partial_{j}+c(x)\) be a second order differential operator. Determine its formal adjoint
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(X_{t}:=\left(B_{t}, \int_{0}^{t} B_{s} d sight)\) is a diffusion process and determine its generator.
Consider the \(\operatorname{ODE} \partial_{t} \xi(t, x)=x+b(\xi(t, x)), \xi(0, x)=x\), where \(b \in \mathcal{C}_{b}^{2}\left(\mathbb{R}^{d}, \mathbb{R}^{d}ight)\) such that
Let \(\left(M^{(1)}, \ldots, M^{(d)}ight)\) be a local martingale with \(M_{0}=0\) and denote its quadratic variation by \(\langle Mangle=\left(\left\langle M^{(j)}, M^{(k)}ightangleight)_{j, k}\).
Let \(\left(N_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a continuous local martingale. Show that \(\tau_{n}:=\left\{s \geqslant 0:\left|N_{t}ight| \geqslant night\}\) is a localizing sequence
Prove the converse of Corollary 23.20.Data From Corollary 23.20 23.20 Corollary (Lvy 1948). Let (Xt, Ft)to be a stochastic process with values in Rd and continuous sample paths. If for all j, k =
Show that there exists a stochastic process \(\left(X_{t}ight)_{t \geqslant 0}\) such that the random variables \(X_{t}\) are independent \(\mathrm{N}(0, t)\) random variables. This process also
Let \(S\) be any set and \(\mathscr{E}\) be a family of subsets of \(S\). Show that\[\sigma(\mathscr{E})=\bigcup\{\sigma(\mathscr{C}): \mathscr{C} \subset \mathscr{E}, \mathscr{C} \text { is
Let \(\emptyset eq E \in \mathscr{B}\left(\mathbb{R}^{d}ight)\) contain at least two elements, and \(\emptyset eq I \subset[0, \infty)\) be any index set. We denote by \(E^{I}\) the product space
Let \(\mu\) be a probability measure on \(\left(\mathbb{R}^{d}, \mathscr{B}\left(\mathbb{R}^{d}ight)ight), v \in \mathbb{R}^{d}\) and consider the \(d\) dimensional stochastic process
Let \(X, Y, X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}, n \geqslant 1\), be random variables. a) If, for all n > 1, Xn Yn and if (Xn, Yn) (X, Y), then XIL Y. b) Let X Y such that X, Y ~ B1/2 = (80
Let \(X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}^{d}, n \geqslant 1\), be two sequences of random variables such that \(X_{n} \xrightarrow{d} X\) and \(X_{n}-Y_{n} \xrightarrow{\mathbb{P}} 0\). Then
Let \(X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}, n \geqslant 1\), be two sequences of random variables.a) If \(X_{n} \xrightarrow{d} X\) and \(Y_{n} \xrightarrow{\mathbb{P}} c\), then \(X_{n} Y_{n}
Let Xn,X,Y:ΩightarrowR,n⩾1Xn,X,Y:ΩightarrowR,n⩾1, be random variables. Ifra \left or missing \ri lim E(f(x)g(Y)) = E(f(x)g(Y)) for all = C(R), g = B(R), n-co
Let \(\delta_{j}, j \geqslant 1\), be iid Bernoulli random variables with \(\mathbb{P}\left(\delta_{j}= \pm 1ight)=1 / 2\). We set\[S_{0}:=0, \quad S_{n}:=\delta_{1}+\cdots+\delta_{n} \quad \text {
Consider the condition for all \(s
Let \(G\) and \(G^{\prime}\) be two independent real N \((0,1)\)-random variables. Show that It: !! G+ G' ~N(0, 1) and . 2
Let \(\left(X_{t}ight)_{t \geqslant 0}\) be a stochastic process on the measure space \((\Omega, \mathscr{A}, \mathbb{P})\) satisfying (B1)(B3). Assume further that there is some measurable set
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\mathbb{P}^{*}(Q):=\inf \{\mathbb{P}(A): A \supset Q, A \in \mathscr{A}\}\) be the outer measure. If \(\Omega_{0} \subset
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. The completion \(\left(\Omega, \mathscr{A}^{\prime}, \mathbb{P}^{\prime}ight)\) is the smallest extension such that
Show that there exist a random vector \((U, V)\) such that \(U\) and \(V\) are one-dimensional Gaussian random variables but \((U, V)\) is not Gaussian.Try \(f(u, v)=g(u) g(v)(1-\sin u \sin v)\)
Let \(G \sim \mathrm{N}(m, C)\) be a \(d\)-dimensional Gaussian random vector and \(A \in \mathbb{R}^{d \times d}\). Show that \(A G \sim \mathrm{N}\left(A m, A C A^{\top}ight)\).
Let \(X, \Gamma \sim \mathrm{N}(0, t)\). Show that - X + I and X ~ N(0, 2t).
Let \(G_{n} \sim \mathrm{N}\left(0, t_{n}ight), n \geqslant 1\), be sequence of Gaussian random variables. Show that \(G_{n} \xrightarrow{\mathrm{d}} G\) if, and only if, the sequence
Let \(A, B \in \mathbb{R}^{n \times n}\) be symmetric matrices. If \(\langle A x, xangle=\langle B x, xangle\) for all \(x \in \mathbb{R}^{n}\), then \(A=B\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Decide which of the following processes are Brownian motions:a) \(X_{t}:=2 B_{t / 4}\);b) \(Y_{t}:=B_{2 t}-B_{t}\);c)
Let \((B(t))_{t \geqslant 0}\) be a \(B M^{1}\).a) Find the density of the random vector \((B(s), B(t))\) where \(0
Find the correlation function \(C(s, t):=\mathbb{E}\left(X_{S} X_{t}ight), s, t \geqslant 0\), of the stochastic process \(X_{t}:=B_{t}^{2}-t, t \geqslant 0\), where \(\left(B_{t}ight)_{t \geqslant
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}, \alpha>0\) and consider the stochastic process \(X_{t}:=e^{-\alpha t / 2} B_{e^{\alpha t}}, t \geqslant 0\).a) Determine
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(\mathscr{F}_{\infty}^{B}=\bigcup_{J \text { countable }}^{J \subset[0, \infty)} \sigma(B(t): t \in J)\).
Let \(X(t), Y(t)\) be any two stochastic processes. Show that (X(u1),..., X(up)) (Y(u),..., Y(up)) for all u 1 implies (X(S1),..., X(Sn)) (Y(t),..., Y(tm)) for all s < < Sm, t < ... < tn, m, n 1.
Let (Ft)t⩾0(Ft)t⩾0 and (Gt)t⩾0(Gt)t⩾0 be any two filtrations and define F∞=σ(⋃t⩾0Ft)F∞=σ(⋃t⩾0Ft) and G∞=σ(⋃t⩾0Gt)G∞=σ(⋃t⩾0Gt). Show that Ft Gt (Vt > 0) if, and
Use Lemma 2.8 to show that (all finite dimensional distributions of) a \(\mathrm{BM}^{d}\) is invariant under rotations. 2.8 Lemma. Let (X+)to be a d-dimensional stochastic process. X satisfies
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion.a) Show that \(W_{t}:=\frac{1}{\sqrt{2}}\left(b_{t}+\beta_{t}ight)\) is a \(\mathrm{BM}^{1}\).b) Are
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion. For which values of \(\lambda, \mu \in \mathbb{R}\) is the process \(X_{t}:=\lambda b_{t}+\mu \beta_{t}
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion. Decide whether for \(s>0\) the process \(X_{t}:=\left(b_{t}, \beta_{s-t}-\beta_{t}ight), 0 \leqslant t
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion and \(\alpha \in[0,2 \pi)\). Show that \(W_{t}=\left(b_{t} \cdot \cos \alpha+\beta_{t} \cdot \sin
Let \(X\) be a \(Q\)-BM. Determine \(\operatorname{Cov}\left(X_{t}^{j}, X_{s}^{k}ight)\), the characteristic function of \(X(t)\) and the transition probability (density) of \(X(t)\).
Show that (B1) is equivalent to Bt-Bs for all 0 < s < t.
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(\tau_{a}=\inf \left\{s \geqslant 0: B_{s}=aight\}\) where \(a \in \mathbb{R}\). Show that \(\tau_{a} \sim \tau_{-a}\) and

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